On a 10x10 "chessboard" a token is placed on the leftmost square of the bottom line a.k.a. a1. You can move said token either one step to the right or one step up within the same column or one step diagonally
(combining right and up).
To make it clear from c4 you may advance in one step to c5, c6 or d6.

How many distinct routes exist to reach the top line's rightmost square (i.e. j10?

There's only one way to start out: in the lower left corner. Expanding outward from there, the number of ways of getting to each square is the sum of the ways of getting to the square to the left, the square diagonally to the lower left, and the square below: